We only show fields where the torsion growth is primitive.
For each field $K$ we either show its label, or a defining
polynomial when $K$ is not in the database.

Additional information

The twist $E^{(-139)}$ of this curve $E$ by $\Q(\sqrt{-139})$ was the earliest elliptic curve proved to have analytic rank $3$, and thus the one first used to give an effective lower bound for the Gauss class number problem. See Proposition 7.4 and Theorem 8.1 in Gross and Zagier's paper in Invent. Math.85 (1986). [The modularity theorem was not yet proved in general, but was known for $E$ by explicit calculation of the modular curve [$X_0(37)$] ; the vanishing of the Heegner trace for discriminant $-139$ was proved by Zagier (Notices of the AMS31 (1984), 739-743).] Soon after Oesterlé obtained several improvements on the constant factor in the bound, one of which was to replace the curve $E^{(-139)}$ of conductor $139^2 37 = 714877$ by the [minimal rank-$3$ curve] of conductor $5077$, whose modularity had recently been proved by Mestre, Oesterlé, and Serre. See pages 34-36 of [Goldfeld's exposition] in the 1985 Bull. AMS.

Data computed by John Cremona, Enrique González Jiménez, Robert Pollack, Jeremy Rouse, Andrew Sutherland and others: see here for details.

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.